Inverse functions

One-to-one functions can be inverted so that the inputs and outputs are reversed. They are crucial where we need effective reversal of processes and calculations, like undoing equations in physics, computer graphics transformations, or decrypting data. Use this resource to learn about inverse functions and how to graph them.

If a function \(f\) maps \(x\) to \(y\), then its inverse function, denoted by \(f^{-1}\), maps \(y\) back to \(x\). For a function to have an inverse, it must be one-to-one. This means that every \(x\) has a unique \(y\).

To inverse a function, we simply swap the \(x\) and \(y\) in each ordered pair. For example:

• Coordinate \((a,b)\) is an element of function \(f\) or \((a,b)\in f\).
• Coordinate \((b,a)\) would be an element of function \(f^{-1}\) or \((b,a)\in f^{-1}\).

The domain and range also swap. So:

• The domain of \(f\) is the range of \(f^{-1}\).
• The range of \(f\) is the domain of \(f^{-1}\).

Graphs of inverse functions

A function \(f\) and its inverse \(f^{-1}\) are symmetrical about the line \(y=x\).

Finding inverse functions

To find the inverse of a function, we simply interchange \(x\) and \(y\), then rearrange the function so that it is in terms of \(y\).

Example 1 – finding inverse functions

For \(f:\mathbb{R}\rightarrow\mathbb{R}\), where \(y=f(x)=\dfrac{x-1}{2}\), find the \(f^{-1}\).

To find the inverse function, we swap or interchange \(x\) and \(y\), then rearrange the function so that it is in terms of \(y\).
\[\begin{align*} y & = \frac{x-1}{2}\\
x & = \frac{y-1}{2}\\
2x & = y-1\\
y & = 2x+1
\end{align*}\]

Therefore, \(f^{-1}(x)=2x+1\).

Exercise

1. Find the inverse of the following one-to-one functions.
1. \(y=x+5\)
2. \(y=4x\)
3. \(y=\dfrac{2x+1}{3}\)
4. \(y=\sqrt{2x-1}\) where \(x\geq\dfrac{1}{2}\)

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